Experimental and numerical investigations of monopile ringing in irregular finite-depth water waves

Experimental and numerical investigations of monopile ringing in irregular finite-depth water waves

Erin E. Bachynski^a,b,∗, Trygve Kristiansen^a, Maxime Thys^b

aNTNU, Department of Marine Technology, 7491 Trondheim, Norway

bSINTEF Ocean (formerly MARINTEK), 7450 Trondheim, Norway

Abstract

In storm conditions, nonlinear wave loads on monopile offshore wind turbines can induce resonant ringing-type responses. Efficient, validated methods which capture such events in irregular waves in intermediate or shallow water depth conditions are needed for design. Dedicated experiments and numerical studies were performed toward this goal. The extensive experimental campaign at 1:48 scale was carried out for Statoil related to the development of the Dudgeon wind farm, and included both a rigid model and a flexible, pitching-type, single degree-of-freedom model.

Twenty 3-hour duration realizations for 4 sea states and 2 water depths were tested for each model. A high level of repeatability in ringing events was observed. Uncertainties in the experimental results were critically examined.

The stochastic variation in the 3-hour maximum bending moment at the sea bed was significantly larger than the random variation in repetition tests, and highlighted the need for a good statistical basis in design. Numerical simulations using a beam element model with a modified Morison wave load model and 2nd order wave kinematics gave reasonable prediction of the ringing response of the flexible model, and of the measured excitation forces on the rigid model in the absence of slamming. The numerical model was also used to investigate the sensitivity of the responses with respect to damping and natural period. A simple single degree-of-freedom model was shown to behave similarly to a fully flexible model when considering changes in natural frequency and damping.

Keywords: ringing; higher order wave loads, hydrodynamic model testing; monopile; offshore wind turbine

1. Introduction

The typical lowest resonance period of large-diameter monopile support structures for offshore wind turbines (OWTs) is 3-5 seconds, which can coincide with the second, third and fourth excitation frequencies of ocean waves in representative storm conditions. Nonlinear high-frequency wave loads can therefore lead to springing- and ringing-type responses. Here, we apply Faltinsen’s definition of ringing: “transient structural deflections at frequencies substantially higher than the incident wave frequencies” [1]. Ringing is thus characterized as a transient event, generally following a high, steep wave, while “springing” is characterized as a steady-state response to sum-frequency wave effects.

There is a significant body of research on ringing and springing loads and responses in deep water wave conditions, particularly related to the construction of tension leg platforms and gravity-based structures for oil and gas in the

∗Corresponding author

Email address: [email protected](Erin E. Bachynski)

1990s. Several theoretical models were developed to describe the nonlinear wave loads which lead to ringing [1–5], and experimental validation was carried out considering both fixed and moving cylinders [6, 7].

In the hydrodynamically intermediate water depths (15-50m) which are relevant for monopile OWTs, the mag- nitude of the nonlinear wave loads is higher than in deep water. Load effects related to steep and breaking waves on a parked turbine may be critical for ultimate limit state (ULS) design, necessitating a better understanding of the load mechanisms and structural responses, as well as improved numerical modeling. The present work describes an experimental and numerical study of ringing of a monopile OWT in intermediate water wave conditions, accounting for natural stochastic variations in the ocean environment. One strength of the present study is the appreciable number of tests: 473 three-hour duration irregular sea states were tested experimentally.

Recently, several experimental studies have been carried out to investigate ULS wave effects on monopile wind turbines, as summarized in Table 1, including the present tests at 1:48 scale with a 6.9 m (full scale) diameter model. The objectives of the experimental campaigns vary somewhat, as do the applied scales and waves.

Table 1: Selected experimental studies of wave effects on monopiles. * indicates that the scale was not explicitly given by the authors, and is expressed here based on a full-size diameter 6.9 m. Wave conditions are classified as DBr: deterministic (focused) breaking waves;

R: regular; I: irregular, LC: long-crested, SC: short-crested. Irregular wave conditions are further classified as FLS (fatigue, non-steep) and ULS (steep).

Facility scale water depth (m) model waves

Wienke & Oumeraci, 2005 [8] FZK 1:9.8* 39.2-44.1* rigid DBr

de Vos et al., 2007 [9] Aalborg University 1:57.5* 20.1-28.7* rigid R, ULS I, LC

Zang et al., 2010 [10] DHI 1:27.6* 13.9 rigid DBr

de Ridder et al., 2011 [11] MARIN 1:30 30 flexible R, DBr, FLS I, LC

Nielsen et al., 2012 [12] DHI 1:36.6 16.6-20.1 rigid ULS I, SC

Nielsen et al., 2012 [12] DHI 1:80 20.8-40.8 rigid/flexible ULS I, SC

Present work NTNU/SINTEF Ocean 1:48 19-27 rigid/flexible ULS I, LC

Wienke and Oumeraci carried out extensive tests with deterministic (focused) breaking waves in order to develop a semi-empirical formulation of breaking wave forces [8]. These tests were carried out with a relatively large scale and depth, and considered only plunging breaking waves. As such, these tests are useful for examining short-duration slamming loads, but do not consider excitation of the lowest global mode of the monopile.

Tests at Aalborg carried out by de Vos et al. examined run-up along the monopile in order to address design concerns related to the placement of access platforms [9]. Although de Vos et al. did not measure loads on the structure, the experimental setup was similar to the present work in terms of scale and tank length. Second order irregular wave kinematics were used to develop run-up prediction formulae. One key difference in the experimental setup is related to the tank bottom: while de Vos et al. applied a ramp (1:100 slope), the waves in the present test were generated at the same depth as the monopile model.

Zang et al. suggested “re-visiting” the ringing phenomenon - known from oil and gas installations in deeper water - in the context of offshore wind turbine foundations in 2010 [10]. Through a limited number of experiments

with focused and inverse focused waves, they proposed to examine the harmonic components of the forcing. While the study was not conclusive, the formulation of the problem in terms of ringing was important, and they observed the importance of second order parasitic waves.

More extensive testing was carried out at MARIN [11]. A 1:30 flexible model was constructed to represent the mass, stiffness, and first two modes of a 6MW wind turbine with 6 m diameter at the waterline (full scale). The focus of the study was on breaking wave impacts, and an extreme case (focused breaking at the monopile) was shown to excite the first three modes of the model and to lead to accelerations at the nacelle of up to 8.9 m/s². Ringing-type responses in irregular wave conditions were also seen to excite the first and second modes of the tower, but no statistical analysis of these observations was shown. The tests at MARIN were carried out with a flat tank bottom.

Researchers in Denmark (DHI/DTU) have also carried out wave tank testing of flexible monopile support struc- tures [12–15]. These tests were carried out in a wave tank with a sloped bottom (1:25) and included regular waves, irregular waves, and focused breaking waves. Short-crested (spread) seas were included in the tests. Significant vibrational responses were observed in steep and in breaking waves. Although the excitation was not distinctly identified as ringing, the results suggested that responses due to steep waves could be significant in short-crested seas (as well as in long-crested seas).

While previous experimental campaigns for monopile wind turbine support structures focused heavily on deter- ministic breaking waves, or on a limited number of events in irregular wave realizations, the present work, which was carried out for Statoil related to the development of the Dudgeon wind farm, makes use of a highly automated test setup such that extensive quantities of data, including numerous repetition tests, have been collected. This improved statistical basis gives new insights in the distribution of maximum loads, both for a rigid and a flexible (single-mode) structure.

In addition to experimental campaigns, numerical studies of monopile foundations have also been carried out.

In order to model the complex flow in the tests at DHI/DTU, two approaches were considered: 1) a fully nonlinear Navier-Stokes volume of fluid approach with loads calculated from pressure integration and 2) Morison’s equation with wave kinematics computed from a fully nonlinear 2D potential flow model of the undisturbed wave field [12, 16, 17]. Both nonlinear solvers are found to work approximately equally well in general, although the Navier-Stokes solver was found to work better for near-breaking waves. Still, both of these methods are relatively computationally expensive and difficult to apply in the ULS design of monopile wind turbines. Since the present experiments employed a flat bottom (no ramp) wave tank, it was decided to estimate the wave kinematics by means of theoretical second order reconstruction. The applicability of a modified Morison’s equation with second order wave kinematics for computing the long-term statistical maximum loads is evaluated in the present work.

The modified Morison’s equation has advantages in terms of simplicity, but there are inherent limitations.

Other methods of accounting for spatial gradients in the undisturbed wave potential have been proposed (i.e.

[5]), but neither these nor the present approach can account for local free-surface diffraction effects. Furthermore, the present work uses the linearized measured wave elevation as input to the numerical model; the ability of the numerical model to recreate the statistical wave process in the absence of measured waves (as would be typical in design) is not considered.

The experimental setup (including an error analysis) is presented in Section 2, while the numerical model is described in Section 3. Results from the experimental and numerical campaigns are presented and discussed in Section 4. The results include time series of selected events, validations of the numerical model, and an analysis of the variation and extreme values of the maximum loads and responses. Furthermore, in order to verify that the experimental findings are relevant for real structures and to extend the applicability of the observed patterns, numerical sensitivity studies were carried out where the responses of a more realistic fully-flexible model were compared to those of the single degree-of-freedom model. Variations in the damping (structural and hydrodynamic) and in the natural frequency were examined.

2. Experimental Setup

The 1:48 scale experiments were carried out in a medium size wave tank at NTNU/SINTEF Ocean (formerly MARINTEK) in Trondheim. The tank length is 28m, the tank breadth is 2.5m and the maximum water depth is 1m. For the present tests, the tank was equipped with a piston-type wavemaker. The experimental set-up is illustrated schematically in Figure 1 and selected video snapshots are shown in Figure 2.

Parabolic beach

w2 w3

w4 w5 w6

w7 w8

w9 w10 w11 w12 w13

w1

Piston wave maker 6.700m

10.182m 13.197m 15.246m (Model: 15.260m) 16.400m – 16.408m

28.0m

2.50m

3.4m 2.5m

Wave dampers

39.6cm (19m) 56.3cm (27m)

Bird's eye view

Side view

. . . .. . . . . . . .. . . .

. . . .. . . . . . . .. . . .

Pit

Concrete tank floor

1.50m

0.8m

1.0m

1.3m

Top mass Acc-meters xand y

Force- and moment transducer

Dry back- side

Leaf spring Monopile model

x z

M_y

Figure 1: Model test set-up and coordinate systems of the present ringing experiments.

2.1. Model and Instrumentation

The particulars of the model are provided in Table 2. The model was constructed by a uniform aluminium pipe with diameter D = 0.144 m (6.9 m full scale) and wall thickness 6mm, a flange at the upper end to support top mass and accelerometers, and a bottom piece at the lower end to fix it to the dynamometer. In the flexible model tests, the model was connected to the force/moment transducer via a leaf spring made by a 5 mm ×50 mm × 50 mm aluminium plate welded to two end pieces (see Fig. 3). The aluminum pipe was relatively rigid, such that the leaf spring provided most of the flexibility and the model had a single pitching-type mode of motion. A top mass was included to mimic the weight of the wind turbine rotor and to achieve the desired natural period. Since

Figure 2: Snapshots from the rear view video. The time in the display is model scale. Upper: non-breaking event. Bottom: severe breaking wave event. Left: a time instant just prior to the event. Right: the time instant where the wave crest aligns with the model.

Time-series of measured waves and mudline overturning moment from the two events are presented in Figure 6.

the maximum height of the model was reduced compared to a full scale wind turbine, the rotor mass was somewhat increased. The natural period in dry condition was approximately 3.7 s (0.53 s or 1.87 Hz model scale).

The model was wet inside, but filled with light-weight porous stuffing in the wetted part to avoid internal sloshing, whose lowest natural periods coincided quite closely with the natural period of the flexible model. Decay tests were performed at 27m water depth without the stuffing, showing that the internal sloshing affected the motions to a limited extent if the stuffing was not used.

In the rigid model tests, the leaf spring was replaced by a cross-bar stiffener, and the top mass was removed.

The natural frequency was then approximately 23 Hz model scale (or 3.3 Hz full scale).

A transparent acrylic glass plate was used to cover the pit (see Figure 2), and care was taken to seal the area where the leaf spring or cross-bar stiffener penetrated the acrylic glass as much as possible to suppress unwanted water motion while still avoiding contact between the model and the acrylic glass plate. The distance between the bottom of the model and the acrylic glass was approximately 3mm, sufficient to avoid contact during the most extreme ringing events.

In reality, monopile structures are flexible with a range of structural modes. The simple single mode model was chosen to reduce the complexity in order to provide benchmark data with as few uncertainties as possible for the numerical work. The single mode model seeks to match the dominant aspects of the first structural mode; the natural period and a representative damping level in the range 0.5 - 2 %.

We acknowledge two main discrepancies between the present, single-mode model and a realistic elastic structure.

The first is that the majority of the motion in reality occurs in the top half of the structure, and the motions in the wet part are modest. The pitching motion of the single mode model includes significantly more motion in the

Figure 3: Force and moment transducer and leaf spring for the flexible model.

wetted part, which causes more wave radiation and viscous damping than a full scale wind turbine. The wave radiation is a dominant part of the total damping in the present tests (see Section 2.2). The second discrepancy is that the higher modes are not represented, and these may account for a non-negligible part of the maximum overturning moment near the base line, especially in wave breaking, or slamming events. In order to examine the effect of the simplified single-mode model on the interpretation of the results, some sensitivity studies were also carried out with a numerical fully-flexible model and compared to results with a numerical single-mode model (see Section 4.3).

Table 2: Model particulars. Top mass includes top flange and accelerometers. Note that the dry resonance period also includes flexibility in the dynamometer.

Model scale (Exp.) Full scale (Num.)

Model height 1.5 m 72 m

Diameter 144 mm 6.9 m

Wall thickness 6 mm 0.288 m

Pipe weight 11 kg 1215 metric tons

Top mass 4.07 kg 409 metric tons

Spring stiffness 2944 Nm/rad 1.56x10⁷ kNm/rad Pitch inertia 18.6 kgm² 4.73x10⁶ tonnes-m²

Resonance period, dry 0.53 s 3.7 s

The instrumentation included a force/moment dynamometer located in the pit just below the tank floor, two accelerometers located at the top of the model (in the horizontalx−andydirections, along-wave and across-wave, respectively), 13 conductance-type wave gauges, two cameras (front and rear-side views), water temperature sensor, and a position-meter to measure the piston type wave flap motion.

2.2. Decay tests

In order to characterize the model, decay tests were carried out by pulling the top of the model and releasing.

Results from decay tests in the dry tank (no water), and 19m and 27m water depths are provided in Figure 4. The maximum amplitude of motion is similar to that seen during ringing events in irregular waves. Time-series of the mudline moment are shown on the left hand side. The lower plots, which all use the same time-window, show that the damping increased with water depth. This is mainly due to wave radiation damping. The critical damping ratio, ζ, obtained by standard logarithmic decay analysis, is shown on the right side of Fig. 4. Linear curve fits to the data are included, where the solid and dashed curves correspond to the peaks and troughs of the decay time-series, respectively. In the 27m water depth case, repetition tests were performed, showing some scatter, but the variation among repetition tests is significantly smaller than the difference in damping due to changes in water depth.

The nonzero slope of the linear fit forζshows that quadratic damping was present in the tests, and its magnitude was similar in all three cases. The origin of the quadratic damping is hypothesized to be related to welds in the leaf spring, because the same quadratic damping could be observed with and without the plexiglass plate and in both wet and dry conditions. There was no evidence of quadratic hydrodynamic damping due to flow separation. This is as expected; the amplitudes of motion in the decay tests (in still water) were small such that the Keulegan-Carpenter number was small, KC . 1.3. Attached boundary layer flow provides non-negligible, but non-dominant linear damping, as shown by [18]. Wave radiation damping was the dominant contributor to linear damping according to our estimates. The wave radiation damping was estimated by integrating panel pressures in WAMIT (see Section 3.2), while the contribution from boundary layer flow was found from the 2-D quadratic damping coefficient [18]:

C_D^F = 4π r π

ReKC, (1)

whereRe is the Reynolds’ number. SinceC_D^F ∝ _U¹ (whereU is the characteristic velocity), a linear 2-D coefficient can be found and integrated along the pitching monopile. The total measured damping is larger than the estimated contributions from structural, wave radiation, and viscous damping; some additional unknown effects (such as the gap between the bottom of the model and the tank floor, or the accelerometer cables) may also play a role.

Table 3: Estimated damping during decay tests, expressed as % critical damping.

19 m 27 m Structural (dry decay) 0.28 0.26 Wave radiation (WAMIT) 0.41 0.98 Attached boundary layer flow 0.01 0.03 Total (wet decay) 0.83 1.30

The waves radiated by the model were reflected back from the wave-maker, as exemplified in the long time series of decay at 27 m water depth in Fig. 4. Approximately 70s after the model was released, the model was re-excited by its own radiated waves, which agrees well with theoretical time to return (69 seconds). The amplitude of the bending moment after re-reflection is approximately 6-7% of the bending moment at the instant of release. This

0 50 100 150 Mudline moment (Nm)-40

-20 0 20

40 _Dry

0 50 100 150

h19

0 50 100 150

h27

Re-reflection

Time (s)

0 10 20

Mudline moment (Nm)-40 -20 0 20

40 _Dry

Time (s)

0 10 20

h19

Time (s)

0 10 20

h27

Mudline moment (Nm)

0 10 20 30 40 50

1

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

Dry 19m

27m

Dry 19m

27m

Dry 19m

27m

Dry 19m

27m

Dry 19m

27m

Figure 4: Decay test results (model scale). Left: Time-series from tests in no water, 19m and 27m water depths. Right: Corresponding logarithmic decay analysis showing critical damping ratioζ. Wave radiation damping is a dominant part of the damping.

corresponds well with the fact that the monopile covers about 6% of the tank width, which is a measure of the model’s ability to absorb the total returning wave energy. We must expect that this self-excitation mechanism has been present during all the irregular wave tests, representing an error source in our experiments, further discussed in Section 2.5.

2.3. Test matrix

Two water depths were chosen for the study: 19m and 27m. For the Dudgeon field, the first is a representative water depth in normal conditions, while the second is a storm surge condition. There are clear finite or shallow water depth effects, particularly at the 19m depth. For this water depth, a significant amount of wave breaking occurred along the tank.

In total, 473 three hour duration (irregular) sea states were tested. In addition, 392 regular wave tests were conducted. The regular wave tests are not discussed in the present work, but treated in [19]. The development of an automated test procedure, by programming the wave machine signal and data recording system, enabled the extensive number of realizations. The automated test procedure had two main benefits. One was that time usage of the test facility was maximized; the tests were run day-and-night, as well as week-ends, with minimal manual labour. The other was that the quality of the test results was enhanced due to the fact that the number of human errors was minimized. Over 12 hours (model scale time), 20 seeds of the same sea state were generated by the wave maker; each seed lasted approximately 26 minutes (3 hours full scale, plus time for transients), with 8 minutes (model scale) pause in between each seed to allow the basin to settle. System checks were performed in between each 12-hour test.

An overview of the test matrix is provided in Table 4. There are four sea states (combinations ofHs,Tp, andγ) and two water depths which are applied to each of the two models. These sea states all represent storm conditions.

The main tests are listed in the first column. Each main test consisted of 20 three hour duration realizations, or seeds, numbered 1 - 20. The next two columns refer to repetition tests. The column denoted by Rep20 indicates whether one repetition of each of the 20 seeds was conducted. The column denoted by Rep5 indicates that seed 10 was repeated 4 or 5 times. The repetition tests were important for studying the level of repeatability in the tests, which was generally found to be quite satisfactory (see also Fig. 6).

The wave conditions in Table 4 are also characterized by a representativeKC and irregular Ursell number (U r) defined by [20]. The representativeKC is calculated as:

KC =2πHs

D

coshkph

sinhkph (2)

whereDis the diameter,k_pis the wave number at the peak frequency, andhis the water depth. This representative KCnumber corresponds to a wave with amplitudeH_s(a rough estimate of the expected maximum) with the velocity according to linear theory at the mean water level. In the crest, the real wave particle velocities may be higher.

The irregular Ursell number is computed as:

Ur = kpHs

2(k_ph)². (3)

and is seen to remain below the typical validity limit for second order waves (U r= 0.33).

2.4. Wave generation and wave spectra

The waves were generated by a piston-type wave maker. The space behind the piston paddle was kept dry by sealing the intersection of the paddle and steel plates on the tank wall with Teflon inserts and an applied overpressure of 1 bar. Linear wave maker theory was applied, so parasitic second order waves were present. Ideally, a second order correction to the wave maker theory should be implemented, for instance as described in [21]. The parasitic waves represent a source of error in these tests.

Examples of measured wave spectra from wave calibration tests for all eight sea state/depth combinations from Table 4 are provided in Figure 5. The thin curves represent the specified spectra, while the measured spectra at the model centerline position are represented by the thick curves. Both the amplitude and phase of the Fourier components were random numbers drawn from Rayleigh and uniform distributions, respectively, according to best practice [22]. The peak frequency 1/T_p (as specified) is indicated by the solid vertical curve in each sub-plot. The dashed curves labeled with numbers 1 - 4 show multiples of the natural period 3.8 s (which corresponds to the first structural natural frequency) to illustrate which wave harmonics are important. For instance, the fourth harmonic of waves around 15.2 s is of importance in the cases withTp= 13.2 s, but not in the other cases.

In the 27 m water depth cases, the whole spectra are quite well reconstructed, whereas in the 19 m water depth cases, it is noticeable that wave energy is lost, in particular in the range 1/Tp.f .2/Tp. This is due to significant wave breaking along the tank under and after large wavemaker motions. No attempt was made to remedy this by boosting the wave maker motion, since more massive wave breaking would then occur. The distribution of wave energy depends on the water depth for these depth-to-wave length ratios (especially at 19 m water depth), and

Table 4: Test matrix. Regular and irregular wave tests. The main tests are listed in the first column. Each main test consisted of 20 three hour duration realizations, or seeds, numbered 1 - 20. The next two columns refer to repetition tests. The column denoted by Rep20 indicates whether one repetition of each of the 20 seeds was conducted. The column denoted by Rep5 indicates that seed 10 was repeated 4 or 5 times. γis calculated as given in DNV RP-C205, except for the broad-band spectrum: Tp= 10s andγ= 1.

Model Water depth Main Rep20 Rep5 H_s T_p γ KC Ur

Flexible

19m

20 20 6.15 10 3.04 7.24 0.16

20 6.15 10 1 7.24 0.16

20 5 6.7 11.5 1.9 8.76 0.21

20 5 6.8 13.2 1 9.94 0.25

27m

20 20 4 6.15 10 3.04 6.46 0.09

20 6.15 10 1 6.46 0.09

20 20 4 6.7 11.5 1.9 7.69 0.12

20 20 4 6.8 13.2 1 8.63 0.14

Rigid

19m

20 5 6.15 10 3.04 7.24 0.16

20 6.15 10 1 7.24 0.16

20 20 5 6.7 11.5 1.9 8.76 0.21

20 5 6.8 13.2 1 9.94 0.25

27m

20 6.15 10 3.04 6.46 0.09

20 6.15 10 1 6.46 0.09

20 20 4 6.7 11.5 1.9 7.69 0.12

20 6.8 13.2 1 8.63 0.14

Sf[m2s]

0 10 20 30 40 50 60 70 80

90 1=T_p

1 2

3 4

h= 19m

8000 Tp= 10s: Hs= 6:15m:.= 3:04

1=T_p

1 2

3 4

h= 19m

8050 Tp= 10s: Hs= 6:15m:.= 1

1=T_p

1 2

3 4

h= 19m

8100 Tp= 11:5s: Hs= 6:7m:.= 1:9

1=T_p

1 2

3 4

h= 19m

8200 Tp= 13:2s: Hs= 6:8m:.= 1

f[Hz]

0 0.1 0.2 0.3

Sf[m2s]

0 10 20 30 40 50 60 70 80

90 1=T_p

1 2

3 4

h= 27m

8506

f[Hz]

0 0.1 0.2 0.3

1=T_p

1 2

3 4

h= 27m

8550

f[Hz]

0 0.1 0.2 0.3

1=T_p

1 2

3 4

h= 27m

8606

f[Hz]

0 0.1 0.2 0.3

1=T_p

1 2

3 4

h= 27m

8704

Figure 5: Examples of irregular wave spectra; seed 10 of each of the eight combinations of sea state and water depth. Measured (thick curves) versus specified (thin curves). The axes are the same for each of the plots. Upper row: h= 19m. Lower row:h= 27m. At 19m water depth, significant energy is lost for wave frequencies 1/Tp.f.2/Tpdue to wave breaking along the tank.

the TMA spectrum could be more realistic than the specified JONSWAP spectrum [23]. Implementation of the TMA spectrum was not attempted here, but could be investigated in further studies. The high frequency tail of the measured waves is quite well reconstructed in the experiments at both water depths, except for the broad-band spectra (Tp= 10s andγ= 1). The low-frequency energy seen in the measured spectra is related to set-down effects.

The set-down (or rise-up) is not expected to provide significant forcing on its own, but can modify the moment arm of higher-frequency loads for individual wave events.

2.5. Assessment of Experimental Error

Experimental error and uncertainty are inherently present in model tests. Although it is not possible to quantify all error effects, it is important to consider finite tank effects, measurement accuracy, and repeatability. Table 5 summarizes the error sources which were examined, and their expected impact on the results.

The finite extent of a wave tank is of general concern because wave energy is not transported away to infinity.

Wave reflections and blockage effects should therefore be considered. In the present test, the ratio between the model diameter and tank breadth was small enough that blockage effects are negligible. This can be seen by considering the potential flow solution of a 2D circular cylinder in infinite fluid, which causes a disturbance of the flow∝(a/r)², where ais the cylinder radius andr is the radial distance from the cylinder center. Here,a/r= 1/17 at the tank

Table 5: Brief summary of experimental error sources. Velocities are given in full scale.

Error source Quantification Comment

Blockage / tank breadth to model diameter ratio 1:17 Negligible blockage (<0.3 %)

Side wall reflections Reflection of high modes

Beach reflections up to 10 % Significant for regular wave tests Self-excitation/ wave maker reflections Appr. 6 % Small effect on maxima

Seiching 0.08 m/s Insignificant

Pumps 0.07 m/s Insignificant

Water depth variation /Refraction <1 % Waves were symmetric

Shear/moment dynamometer 1 % Static tests

Wave gauges 2 % + 2 %/deg C Corrected for temperature

wall. The effects of reflections (of diffracted waves) from the side walls are less clear. The first order diffraction is negligible due to the long wave regime for the majority of the wave frequency content, but second and higher order diffraction may be significant, and may be affected by reflections from the side walls.

Attempts were made to limit reflections from the beach. The beach was of parabolic shape, consisting of 4 % perforated plates fixed to a rigid framework which was fastened to the tank floor. A thorough reflection analysis was not performed, but a reflection level of about 10 % forT >12.5s, and less than 3 % forT ≤12.5 was estimated based on regular wave tests.

Self-excitation - waves generated by the model and reflected back from the wave maker - had a measurable effect.

This was illustrated by the decay test at 27 m water depth presented in Figure 4. It is not easy to state exactly how much impact this had on the results. If we consider the maximum overturning moment at mudline during a ringing event, the response depends on the ringing wave load itself and the state of the system as the wave enters, i.e. the underlying springing response. In terms of magnitude, the maximum bending moments after the major ringing events are typically about three times higher than those caused by the underlying springing motion. Based on the model’s ability to absorb approximately 6 % of the re-reflected energy, and that the reflected energy is primarily due to springing events, re-reflection could account for about 2 % of the maximum response magnitude. Despite the limited magnitude of the loads due to re-reflection, their effect on the phasing of the underlying springing motion may nonetheless have a greater impact on the ringing responses (see Section 4.1). For tests with a more realistic mode shape, the self-excitation would be less pronounced: the motion of the wet part would be significantly smaller, with consequently less radiated waves.

Seiching corresponds to longitudinal standing waves in the tank. The periods of the two first modes are 2L/√ gh and L/√

gh, where Lis the length of the tank. This means 56s - 160s (full scale). A slowly oscillating current is associated with the seiching motion. With the observed seiching wave amplitudes, these corresponded to current amplitude of up to 0.08 m/s (full scale) at the model position. Compared to typical wave particle velocities in extreme events exceeding 3-4 m/s, this is small. An additional, unrelated current was also present due to leakage behind the wavemaker piston flap and corresponding pumping of water. The volume flux corresponded to

approximately 1 mm/s model scale, averaged over the tank cross-sectional area. This corresponds to 0.07 m/s full scale, which is expected to give similarly small effects.

Shallow water waves are sensitive to transverse variations in the water depth due to refraction. In order to limit refraction effects, the tank floor was levelled prior to the experiments. The variation in water depth was less than 1 mm in the transverse direction throughout the tank, and 1-2 mm in the longitudinal direction in the last 2/3 of the tank (the 1/3 closest to the wave maker was approximately 3-4 mm deeper). No evident signs of refraction were observed visually or by the measurements (wave gauge pairs).

A significant effort was made in keeping measurement errors low. The shear/moment dynamometer was cali- brated before the tests, and checked regularly during and after testing. Static tests over the course of the exper- imental campaign showed that the variability in base shear and overturning moment measurements were within 1%. Conductance-type wave gauges were used, and were calibrated to absolute variability (maximum deviation between two wave gauges) within 2% for small-amplitude linear waves before the tests. The agreement for larger wave amplitudes is significantly better. Due to temperature variations in the tank, corrections to the wave gauge measurements were applied in post-processing of the data by using the mean measured temperature over each sea state.

2.6. Repeatability

Finally, repeatability will be considered in greater detail. Regarding the wave maker motion, each wave was run at least three times: wave calibration test, rigid model test and flexible model test. The absolute deviation of the wave maker motion between tests was less than 0.4 %.

Time-series from one major ringing event and one breaking wave event are provided in Fig. 6, illustrating the typical level of repeatability of the loads. The time series correspond to the video snapshots in Fig. 2. Results from the flexible model test (response) and corresponding rigid model test (load) are presented together in Fig. 6.

The tests were repeated five times, meaning that results from twelve runs are presented. The deviation of the peak mudline moment is within 1-2 % in the (representative) non-breaking case, while breaking wave events are less repeatable. The deviation of the mudline moment is around 8-10 % in the breaking case. The repeatability level for the non-breaking event (top) is representative of the major part of the tests.

With the flexible model, there are five combinations of wave condition and water depth where seed 10 was repeated 6 times in total. The variation in the maximum wave elevation, shear force, and mudline moment is summarized in Table 6. The coefficient of variation is computed asσ/µ, whereσis the standard deviation of the 6 maxima andµis the mean of the 6 maxima.

3. Numerical model

A numerical model of the experimental setup at full scale was created in SINTEF Ocean’s nonlinear beam finite element code RIFLEX.

3.1. Structural model

The RIFLEX model included beam elements which were used to represent the dynamometer, the single element leaf spring, the wet part of the monopile, and the dry part of the monopile. The internal fluid (water) was included

4640 4650 4660 4670 4680 4690 4700 4710 4720 4730 time [s]

-25 -20 -15 -10 -5 0 5 10 15 20

wave w7 [m]

My, flexible [x10⁴ kNm]

My, rigid [x10⁴ kNm]

4645 4650 4655 4660

time [s]

-5 0 5 10 15 20

(Steep, non-breaking wave)

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 time [s]

-25 -20 -15 -10 -5 0 5 10 15 20 25

wave w7 [m]

My, flexible [x10⁴ kNm]

My, rigid [x10⁴ kNm]

1900 1905 1910 1915

time [s]

-5 0 5 10 15

20 (Breaking wave event)

Figure 6: Illustration of typical repeatability level in the tests. Hs = 6.7 m, Tp = 11.5 s, γ= 1.9, water depth h= 27 m. Steep, non-breaking ringing event (top) and breaking wave event (bottom). The figures to the right are zoomed views of the left figuresMy

indicates overturning moment at the dynamometer. Six tests with flexible model and six tests with rigid model are shown.

in the mass model of the wetted elements. Parameters describing the structural model and discretization are given in Table 7.

In Table 7, note that the Young’s modulus for aluminum at model scale is 65 GPa, but that this corresponds to a higher stiffness at full scale. Since the computations are intended to reflect the model at “full scale,” the scaled-up stiffness must be used. The coating in the experiment was paint. An additional point mass (451 tonnes) was used to represent the top mass. The flexibility in the dynamometer was represented by a flexible joint, with rotational stiffness 6.245e+06 kNm/deg.

Structural damping was applied as stiffness-proportional Rayleigh damping, determined based on the decay test of the structure in air, without top mass. The structural damping coefficient wasa2= 0.00125s, which corresponds to 0.5% of the critical damping when the top mass is present.

Table 6: Coefficent of variation, flexible model tests with 6 repetitions.

Water depth (m) H_s (m)/T_p (s)/γ σ_ηmax/µ_ηmax σ_Fxmax/µ_Fxmax σ_Mymax/µ_Mymax

19 6.7/11.5/1.9 0.0229 0.0439 0.0574

6.8/13.2/1.0 0.0202 0.0128 0.0244

27

6.15/10.0/3.04 0.0221 0.0093 0.0138

6.7/11.5/1.9 0.0135 0.0321 0.0148

6.8/13.2/1.0 0.0127 0.0123 0.0318

Table 7: Structural model in RIFLEX.

Line Length (m) Number of elements Cross-section

dynamometer 8.64 6 D= 3 m,t= 0.1 m,E= 2100 GPa

spring 2.4 6 2.4 m x 0.384 m,E= 3312 GPa

monopile (wet) h 19 or 27 D= 6.912 m,t= 0.288 m, coating 0.044 m,E= 3312 GPa monopile (dry) 72.0−h 45 D= 6.912 m,t= 0.288 m, coating 0.044 m,E= 3312 GPa

3.2. Wave load model

The hydrodynamic loading in the numerical model is based on a modified Morison model, applied to each strip of the monopile:

dFh=ρπD²

4 dzah+ρπD²

4 dzCa(ah−x) +¨ ρ

2DdzCD(uh−x)|(u˙ h−x)| −˙ ρp

gDD²CLdzx,˙ (4) wherea_h andu_h are the undisturbed wave particle total acceleration (including the advective terms) and velocity, respectively. For the acceleration, we use:

a_h= ∂u

∂t +u∂u

∂x +v∂u

∂y +w∂u

∂z (5)

(the transverse velocityvis zero for long-crested waves). Furthermore, in Eq. 4,ρis the density of fresh water,Dis the mean diameter of a vertical strip of the monopile,Ca is the so-called mass coefficient,CDis the quadratic drag coefficient, andCL is a linear damping coefficient. The structural velocity is ˙xand the structural acceleration is ¨x.

The first two terms on the right hand side are usually referred to as the inertia terms, the third as the quadratic drag, or viscous, term, and the final term is a linear damping term used to account for the wave radiation damping at the natural period. Note that only the local acceleration is included in the standard Morison’s equation. We therefore refer to the presently used load model as amodified Morison model. Our model differs from the FNV [24]

model in that the term u^∂u_∂x is included in the added mass term, and the third order diffraction term is omitted.

Due to the relatively large motions of the single degree-of-freedom model at the waterline, the linear damping term was found to be important.

In all of the simulations with irregular waves, we have taken CD= 0.9 andCa = 0.772 and applied a time step dt= 0.1 s. Both coefficients are reasonable average values based on the KC and Reynolds numbers of steep ULS wave events [25]. More detailed estimates ofCD andCa, depending on KC and Reynolds number for each sea state

and depth, could also be considered, but this simplified approach was found to be sufficient for investigating the higher-harmonic load components. The value of CL varied with depth in order to mimic the radiation damping in the system. The stripwise radiation damping coefficients were computed using integration of panel pressures from the radiation potential flow solution, acquired by using WAMIT, at the natural frequency of the model. Two potential flow solutions were considered - one for surge motion, and another for pinned rotation about a point at the seabed. The two approaches gave similar results for the depth-dependent sectional linear damping coefficients (for transverse element motion). Fig. 7 compares the stripwise damping coefficients obtained by each method.

0 0.05 0.1 0.15 0.2

C_L

-30 -25 -20 -15 -10 -5 0

z, m

WAMIT, 19m, surge WAMIT, 19m, pinned rot.

19m, applied in RIFLEX WAMIT, 27m, surge WAMIT, 27m, pinned rot.

27m, applied in RIFLEX

Figure 7: Linear damping coefficientCL, as calculated in WAMIT and as applied in RIFLEX (see Eq. 4).

When combined with the structural damping coefficients from the dry decay tests, the coefficients obtained in WAMIT were found to overpredict the damping in the wet decay tests, and the sectional coefficients applied in the RIFLEX model were therefore scaled down consistently from the obtained coefficients, as shown in Fig. 7. For 19 m water depth, a factor of 0.45 was used for the surge coefficients, while for 27 m, a factor of 0.8 gave the best match.

The reason for this discrepancy is not understood.

The simulations were carried out with three different wave kinematics models, as described in Table 8, in order to see the effects of the different load components. The 2nd order wave formulation follows [26]. If we consider only the terms in Eq. 4 which are related toah, the R0 model can be considered as a purely linear model, while the R1 model captures some 2nd order effects due to the integration of the loads up to the linear free surface. In the R1 model, the kinematics above the free surface are taken to be those ofz= 0. The R2 model includes some 3rd and 4th order effects due to the integration of second order kinematics up to the 2nd order wave elevation. Keeping second order components in the Taylor expansion, the wave kinematics abovez= 0 are extrapolated from the first order potential. As indicated in Fig. 5, wave loading at frequencies at twice and three times the primary wave frequencies is important due to excitation the natural frequency of the model. Due to the viscous drag term, the R0 model includes some forcing at three times the wave frequency which can excite the model. For the given conditions, however, the loads are inertia-dominated. The steepness of the wave during the zero upcrossing (associated with high horizontal wave particle accelerations) is therefore expected to be important for the structural response.

It should be noted that the present modified Morison’s equation is a significant simplification of the real physical problem. First order near-field diffraction effects are not expected to be important for the present situation; the importance of 2nd and higher order diffraction (which is not modelled) is not well known. For larger oil and gas structures in deep water, ignoring the 2nd order diffraction terms is known to result in overestimation of steady-

Table 8: Load models

Model Description

R0 1st order wave kinematics, integration to z = 0 (original Morison model)

R1 1st order wave kinematics, integration to undisturbed 1st order wave elevation (constant kinematics abovez= 0) R2 2nd order wave kinematics, integration to undisturbed 2nd

order wave elevation (modified Morison model)

state springing loads [28]. The regular wave tests in the present campaign also suggest that the same overprediction occurs for the 2nd and, in particular, the 3rd harmonic for similar wave conditions and water depths [19]. The present work compares this simplified method against experimental results; more refined load models which can more correctly model the wave-structure interaction should be considered in future work.

The same linearized measured wave elevation (in the absence of the model) was given as input to RIFLEX regardless of the wave kinematics model. The linearization process attempts to remove second- and higher-order components of the wave elevation from the signal, while keeping the linear components in the tail of the spectrum.

The cut-off frequency and roll-off in the applied low-pass filter depend on the wave conditions and water depth, and are iteratively chosen based on the second order reconstruction of the wave elevation. An example of the measured, linearized, and second order reconstructed wave elevation is shown in Fig. 8. In this example, the cut-off is near 0.13 Hz, which is significantly lower than the deep water recommendationω=p

(2g/Hs) given by [27]. The results of the response analysis are sensitive to the choice of cut-off and roll-off in the linearization.

0 0.05 0.1 0.15 0.2 0.25 0.3 f [Hz]

0 10 20 30 40 50 60 70

Sf [m2s]

linearized reconstructed measured

Figure 8: Measured, linearized, and reconstructed wave spectra for one realization withHs6.7 m,Tp11.5 s,γ1.9, water depth 19 m.

4. Results

As described in Table 4, experiments were carried out for 20 three-hour duration realizations of four sea states for two physical models and two water depths, corresponding to a total of 320 basic cases. Each of these cases was also simulated numerically using three models of the wave kinematics, such that 960 simulations were carried out. In this section, we present and discuss the main results. General trends and selected events are examined by means of examples of time-series comparisons of simulations and experiments in Section 4.1. The variation and distribution of maximum responses are discussed in Section 4.2. Lastly, numerical sensitivity studies related to the level of structural and hydrodynamic damping and to the natural frequency of the model are presented in Section 4.3.

4.1. Time series examples

A full 3-hour experimental realization is illustrated in Fig. 9, which includes the waiting period after the con- clusion of that realization. These time series, as well as the corresponding spectra provided in Figure 10, highlight the response of the single-mode model at its natural frequency. In addition to the dynamic amplification related to ringing events, there is also a significant underlying, springing-type, response ahead of the events. The underlying motion has consequences for the maximum loads, through the phasing with the ringing load. This naturally applies to ringing of all kind of structures, but is perhaps more evident here since the underlying motion has a quite large amplitude relative to those at maximum loads. If the pitching motion of the model is a minimum (pitched towards the wave) when the load is at its maximum, the underlying motion will add to the maximum load. We refer to this as additive or unfavorable phasing. In the other extreme case, the phasing non-additive, or favorable for the structure, since it will cause the maximum response to be lower than if the model was at rest or in any other state.

In the two examples provided in Figure 6, the upper is additive while the lower is non-additive.

Four selected ringing events are examined here in order to study the behavior of the model, and in order to compare the experimental and simulation results in detail. The wave elevation, shear force, and overturning moment at the dynamometer for the selected events are shown in Fig. 11-14. Results from the rigid model, which can be interpreted as the external hydrodynamic loads, and from the flexible model, including the dynamic response, are shown. When available, experimental results from repetition tests are included.

Fig. 11 exemplifies the results from 19 m water depth for a non-breaking, steep wave event. By comparing the calibrated wave elevation with the simulation results, one can clearly see that the 2nd order theory is an improvement over the first order theory in terms of the crest height and steepness. Still, the physical wave in the basin contained more nonlinearity than a 2nd order model was able to capture, as illustrated by the higher crest of the measured wave. The 2nd order (R2) model underestimates the maximum hydrodynamic force and moment acting on the rigid monopile in the given example by about 15 %. Despite the underestimation of the maximum hydrodynamic loads, the 2nd order model was able to predict theresponse of the flexible model fairly well (10 % overprediction). The R1 and R2 models both captured some of the resonant response, while - as expected - the R0 (linear wave kinematics and integration to the mean free surface) model showed primarily wave-frequency response.

The R2 model captured the ringing response more accurately than the R1 model, but the springing response prior

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-2 -1 0 1 2 3

M y [kNm]

#10⁵

4600 4700 4800 4900 5000 5100

time [s]

-3 -2 -1 0 1 2 3

M y [kNm]

#10⁵

Ringing event

Figure 9: Examples of measured mudline overturning moment (same test as Fig. 6). Full scale values are shown. Upper: complete three-hour realization including waiting after the test. Lower: zoom-in around the non-breaking ringing event.

0 0.05 0.1 0.15 0.2 0.25 0.3 f [Hz]

0 2 4 6 8

S

[(kNm)

s]

#10

My, flexible My, rigid

Figure 10: Power spectra of the overturning moment at the dynamometer for the rigid and flexible tests, showing the strong dynamic amplification at the natural frequency.

to the large wave passage was overpredicted by the R2 model. For the given wave condition withTp= 11.5 s, the 3ω(3rd harmonic) loads are expected to have dominated the response.

-4 -2 0 2 4 6

2 [m]

R0 R1 R2 Exp

-2000 0 2000 4000

F x (rigid) [kN]

-4000 -2000 0 2000 4000

F x (flex) [kN]

-4 -2 0 2 4 6 8

My (rigid) [kNm]

#10⁴

1835 1840 1845 1850 1855 1860 1865 1870 t [s]

-1 0 1

My (flex) [kNm]

#10⁵

Figure 11: Time series of the calibrated wave (top), shear force (two and three) and overturning moment (four and five). Results for the rigid and flexible structure are shown in separate subplots. The time window corresponds to the largest bending moment for the flexible model from one of the experimental realizations withHs6.7 m,Tp11.5 s,γ1.9, water depth 19 m. Additive phasing. Measured results from the experiment are denoted Exp.

Fig. 12 shows an example of the same wave condition as in Fig. 11 but for 27 m water depth. Compared to the results for 19 m, the R2 model captured the maximum hydrodynamic loads on the monopile relatively well for 27 m, although the maximum wave elevation was still underpredicted. The maximum load was here slightly overpredicted by the R2 model. Good agreement between the R2 model and the experiments is seen in the maximum dynamic response. The amplitude of the springing response prior to the event was captured by the R2 and R1 models, and the phase of the springing response is in good agreement with the experiment for the cycles prior to the passage

of the large wave. A general trend is that the numerical model (R2) is in better agreement with the experiments at 27 m than at 19 m water depth. We believe that this can - to a certain extent - be explained by the fact that the 2nd order wave model is partly outside its range of applicability at 19 m (and sometimes also at 27 m) for our considered cases. Furthermore, the wave nonlinearity is expected to be more severe in 19 m than in 27 m water depth.

The limitations of the applicability of the 2nd order wave kinematics can be seen by examining the wave elevation time series R2 in Figs. 11 and 12. In the 19 m water depth (Fig. 11), a non-physical “bump” in the R2 wave elevation can be seen around 1850s, indicating that the second order theory is not valid. The same wave condition in 27 m water depth in Fig. 12 does not reveal similar “bumps”.

-5 0 5

2 [m]

R0 R1 R2 Exp

-5000 0 5000

F x (rigid) [kN]

-5000 0 5000

F x (flex) [kN]

-1 0 1 2

My (rigid) [kNm]

#10⁵

9865 9870 9875 9880 9885 9890 9895 9900 t [s]

-2 -1 0 1 2

My (flex) [kNm]

#10⁵

Figure 12: As in Fig. 11 forHs 6.7 m,Tp11.5 s,γ1.9, water depth 27 m. Additive phasing.

The importance of the springing response on the ringing response is exemplified in Fig. 13. Comparing Figs. 12

and 13, the maximum wave elevation and hydrodynamic loads on the rigid cylinder can be seen to be similar. The numerical simulations with R2 are seen to capture the hydrodynamic loading quite well. The experimental results show, however, that the dynamic response following the steep wave in Fig. 13 was much smaller than in Fig. 12. In the event in Fig. 12, the structure was already accelerating in the same direction as the maximum hydrodynamic load at the time of the maximum hydrodynamic load (additive phasing). This can be seen in the last two subplots:

at the time instant where the maximum wave-induced moment occurred, the high-frequency part of the bending moment in the flexible structure was already increasing. In Fig. 13, the springing-induced moment had already reached its maximum value and was decreasing at the time when the maximum hydrodynamic moment was applied (non-additive). The subsequent response was approximately 40 % smaller than in Fig. 12. Furthermore, the R2 numerical simulation, which didn’t match the phase of the springing response in Fig. 13, is seen to underpredict the response for that particular event. The reason for the discrepancy in the underlying springing motion is not understood. Beach reflections, re-reflections, parasitic second order waves, and higher order diffraction can all contribute to differences in the springing response.

Finally, Fig. 14 shows an extreme event in 27 m water depth with slamming-type loads from a breaking wave.

As shown, the measured wave-induced moment on the rigid cylinder was slightly larger than the corresponding positive bending moment of the flexible model. The subsequent negative bending moment for the flexible structure was, however, significantly higher. This case also illustrates the challenges encountered when measuring the loads on the rigid model: the experiments were designed to capture the ringing excitation from steep, non-breaking waves, but the system was not sufficiently stiff to accurately capture the high-frequency peaks in slamming-type loads:

the natural frequency was 23 Hz (model scale). The results shown in the present work include a filter at 10 Hz model scale. Fig. 15 exemplifies the effects of this filter for a breaking wave event. In Fig. 14, one can also see a non-physical bump in the wave elevation at 4602 s, indicative of exceeding the limits of second order wave theory, even at 27 m, due to the steepness of this event.

To summarize, a few general observations can be made from the time series of results, such as those shown in Fig. 11-14:

1. The global hydrodynamic loads were observed to be inertia-dominated, as can be expected with the governing low KC numbers (see Table 4), and a maximum positive load occurred approximately at the part of the wave time-series with the fastest rise-time, where the horizontal acceleration (of the undisturbed wave) was largest.

Locally, viscous effects may be important, especially since the estimated KC numbers do not account for nonlinearity in the wave kinematics.

2. The 2nd order wave model (R2) approximated the wave elevation of steep wave events significantly better than the first order models (R0 and R1), although the maximum crest heights in general were underpredicted compared to experiments. This is as expected, since the experimental waves contain more nonlinearity than that predicted by 2nd order wave theory.

3. There was significant dynamic amplification: the flexible model experienced larger load effects than the measured loads on the rigid model, typically about 3 times larger in non-breaking ringing-type events.

4. There was an underlying springing (steady-state) response present in both the experiments and the R1 and R2 simulations. This was observed across all of the tests. The R2 simulations tended to overestimate the

springing responses, especially in the 19 m water depth. The springing response had a noticeable effect on the maxima of the ringing response. Re-reflections (from waves generated by the flexible model) are considered to be a relevant error source in this respect, and the simplicity of the load model (which does not account for first or second order near-field diffraction) may be a reason for overestimation.

5. Ringing-type responses were seen after the passage of a large, steep wave. The maximum absolute value of the response was often in the negative direction (after the initial large positive load). The ringing response decayed slowly, “riding” on top of the first order response. The magnitude of the ringing response was typically best estimated by the R2 model, but was not consistently over- or under-estimated (see also the previous point and Section 4.2).

6. Although not shown explicitly in the selected time series results, the maximum response of the flexible model did not necessarily correspond to the maximum load on the rigid model. Slamming-type peak loads from breaking waves were generally larger than steep wave loads, but the flexible structure exhibited larger response to steep non-breaking waves, as the slamming loads were of very short duration compared to the structural natural period. Slamming loads may become more important if a fully-flexible model is considered [29].

7. The second order wave model was observed to be outside its range of validity for certain long and steep wave events, particularly at the 19 m water depth.

-4 -2 0 2 4

2 [m]

R0 R1 R2 Exp

-2000 0 2000

F x (rigid) [kN]

-4000 -2000 0 2000 4000

F x (flex) [kN]

-5 0 5

M y (rigid) [kNm]

#10⁴

5725 5730 5735 5740 5745 5750 5755 5760 t [s]

-2 -1 0 1

M y (flex) [kNm]

#10⁵

Figure 13: As in Fig. 11 forHs 6.7 m,Tp11.5 s,γ1.9, water depth 27 m. Non-additive phasing.

-5 0 5 10

2 [m]

R0 R1 R2 Exp

-4000 -2000 0 2000 4000

F x (rigid) [kN]

-5000 0 5000

F x (flex) [kN]

-5 0 5 10 15

M y (rigid) [kNm]

#10⁴

4575 4580 4585 4590 4595 4600 4605 4610 t [s]

-2 -1 0 1 2

M y (flex) [kNm]

#10⁵

Figure 14: As in Fig. 11 forHs6.8 m,Tp13.2 s,γ1, water depth 27 m. Slamming event.

1885 1890 1895 1900 1905 1910 1915 1920 1925

time [s]

-1 -0.5 0 0.5 1 1.5 2

My [kNm]

#10⁵

raw filtered

Figure 15: Raw and filtered bending moment for the rigid model, breaking wave event from Fig. 6.